The main goal of this paper is to compute two related numerical invariants of a primitive ideal in the universal enveloping algebra of a semisimple Lie algebra. The first one, very classical, is the Goldie rank of an ideal. The second one is the dimension of an irreducible module corresponding to this ideal over an appropriate finite W-algebra. We concentrate on the integral central character case. We prove, modulo a conjecture, that in this case the two are equal. Our conjecture asserts that there is a one-dimensional module over the W-algebra with certain additional properties. The conjecture is proved for the classical types. Also, modulo the same conjecture, we compute certain scale factors introduced by Joseph, this allows to compute the Goldie ranks of the algebras of locally finite endomorphisms of simples in the BGG category . This completes a program of computing Goldie ranks proposed by Joseph in the 80's (for integral central characters and modulo our conjecture). We also provide an essentially Kazhdan-Lusztig type formula for computing the characters of the irreducibles in the Brundan-Goodwin-Kleshchev category for a W-algebra again under the assumption that the central character is integral. In particular, this allows to compute the dimensions of the finite dimensional irreducible modules. The formula is based on a certain functor from an appropriate parabolic category to the W-algebra category . This functor can be regarded as a generalization of functors previously constructed by Soergel and by Brundan-Kleshchev. We prove a number of properties of this functor including the quotient property and the double centralizer property. We develop several side topics related to our generalized Soergel functor. For example, we discuss its analog for the category of Harish-Chandra modules. We also discuss generalizations to the case of categories over Dixmier algebras. The most interesting example of this situation comes from the theory of quantum groups: we prove that an algebra that is a mild quotient of Luszitg's form of a quantum group at a root of unity is a Dixmier algebra. For this we check that the quantum Frobenius epimorphism splits.
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